unless you open a cs2 file (in cs6) and save it using cs6 format, cs2 should be able to open it.
As soon as I installed CS6, all my CS2 files had a CS6 icon next to them. I then opened CS2 files within CS2 (and saved them in CS2 format) and it is very glitchy. When I click on a text block to move it the text disappears.
<moving to indesign by mod - kglad>
you see the CS6 icon, but the file is still CS2, as kglad says, until opened and saved in CS6. I have CS2, CS4 and CS6 installed and working fine.
what OS you have?
Did you maybe have CS2 installed on Win7? And then you installed CS6 on the same machine? If so...
CS2 on Win7 is known to be flaky, and since versions of InDesign sometimes share files, it's awfully easy to accidentally mess up a Really Old Install (like, say, CS2) by installing something newer next to it.
If my guess is correct, and if you need to keep CS2 available, I'd suggest something drastic: Uninstall everything. Make a virtual machine of XP. Install CS2 there. Install CS6 in Win7. That keeps 2005 and your 2013 separated, which I've found (as someone who still works in CS and CS2 and Pagemaker periodically) to be really important.